Question

which of these relations are functions? select all that apply.

Explanation:

Step1: Recall the vertical line test

A relation is a function if no vertical line intersects its graph more than once.

Step2: Analyze the first graph (top - left)

Draw a vertical line (mentally) through the graph. It will intersect the graph at more than one point (the graph is a curve that crosses a vertical line twice), so it is not a function.

Step3: Analyze the second graph (top - right, circle)

A vertical line through the circle will intersect it at two points (e.g., at \(x = 0\), the vertical line intersects the circle at two \(y\) - values), so it is not a function.

Step4: Analyze the third graph (bottom - left, line)

For a linear graph (a straight line with a non - vertical slope), any vertical line will intersect it at most once. So this graph represents a function.

Step5: Analyze the fourth graph (bottom - right, curve)

For this curve, any vertical line will intersect it at most once. So this graph represents a function.

Answer:

The bottom - left graph (the linear graph) and the bottom - right graph (the curve) are functions.